In the stable regime (heavier liquid is below the lighter liquid), local well-posedness and the global well-posedness with constraints on the initial data have been widely studied, with the lowest space \(H^{\frac{3}{2}}\)  ([1]},[2]},[3]},[4]},[5]},[6]},[7]},[8]},[9]},[10]},[11]},[12]},[13]},[14]},[15]},[16]},[17]},[18]},[19]},[20]},[21]},[22]}, [23]}). The existence of self-similar solutions has also been proved [24]}. Interesting readers can see [24]} and [19]} for detailed reviews. Due to the parabolic behavior, instant analyticity has been proved in the stable regime. Castro–Córdoba–Fefferman–Gancedo–López-Fernández [27]} proved the \(H^4\)  solutions become instantly analytic if the solutions remain to be in the stable region for a short time. In [15]}, also in the stable region, Matioc improved the instant analyticity to \(H^{s}\) , where \(s\in (\frac{3}{2},3)\) . In [29]}, Gancedo–García-Juárez–Patel–Strain showed that in the stable regime, a medium size initial data in \(\dot{\mathcal {F}^{1,1}}\cap L^2\)  with \(\Vert f\Vert _{\mathcal {F}^{1,1}}=\int |\zeta ||\hat{f}(\zeta )|d\zeta \)  becomes instantly analytic. Their result also covers the different viscosities case and the 3D case.
